Two families of Exel-Larsen crossed products

TL;DR

This paper studies two families of Exel-Larsen crossed products, analyzing their structure, simplicity, and pure infiniteness, with applications to compact abelian groups and 2-graph algebras.

Contribution

It extends Exel's crossed product construction to new settings, proving a uniqueness theorem and identifying conditions for pure infiniteness and simplicity.

Findings

Crossed products are purely infinite simple for certain connected compact abelian groups.

The natural actions of semigroups on UHF cores exhibit properties similar to single endomorphisms.

A uniqueness theorem for the crossed product is established.

Abstract

Larsen has recently extended Exel's construction of crossed products from single endomorphisms to abelian semigroups of endomorphisms, and here we study two families of her crossed products. First, we look at the natural action of the multiplicative semigroup $\mathbb{N}^\times$ on a compact abelian group $\Gamma$, and the induced action on $C(\Gamma)$. We prove a uniqueness theorem for the crossed product, and we find a class of connected compact abelian groups $\Gamma$ for which the crossed product is purely infinite simple. Second, we consider some natural actions of the additive semigroup $\mathbb{N}^2$ on the UHF cores in 2-graph algebras, as introduced by Yang, and confirm that these actions have properties similar to those of single endomorphisms of the core in Cuntz algebras.